Paola Bortot

Inference for Stereological Extremes

In the production of clean steels, the occurrence of imperfections—so-called “inclusions”—is unavoidable. The strength of a clean steel block is largely dependent on the size of the largest imperfection that it contains, so inference on extreme inclusion size forms an important part of quality control. Sampling is generally done by measuring imperfections on planar slices, leading to an extreme value version of a standard stereological problem: how to make inference on large inclusions using only the sliced observations. Under the assumption that inclusions are spherical, this problem has been tackled previously using a combination of extreme value models, stereological calculations, a Bayesian hierarchical model, and standard Markov chain Monte Carlo (MCMC) techniques. Our objectives in this article are twofold: (1) to assess the robustness of such inferences with respect to the assumption of spherical inclusions, and (2) to develop an inference procedure that is valid for nonspherical inclusions. We investigate both of these aspects by extending the spherical family for inclusion shapes to a family of ellipsoids. We then address the issue of robustness by assessing the performance of the spherical model when fitted to measurements obtained from a simulation of ellipsoidal inclusions. The issue of inference is more difficult, because likelihood calculation is not feasible for the ellipsoidal model. To handle this aspect, we propose a modification to a recently developed likelihood-free MCMC algorithm. After verifying the viability and accuracy of the proposed algorithm through a simulation study, we analyze a real inclusion dataset, comparing the inference obtained under the ellipsoidal inclusion model with that previously obtained assuming spherical inclusions.

The multivariate Gaussian tail model: an application to oceanographic data

Optimal design of sea-walls requires the extreme value analysis of a variety of oceanographic data. Asymptotic arguments suggest the use of multivariate extreme value models, but empirical studies based on data from several UK locations have revealed an inadequacy of this class for modelling the types of dependence that are often encountered in such data. This paper develops a specific model based on the marginal transformation of the tail of a multivariate Gaussian distribution and examines its utility in overcoming the limitations that are encountered with the current methodology. Diagnostics for the model are developed and the robustness of the model is demonstrated through a simulation study. Our analysis focuses on extreme sea-levels at Newlyn, a port in south-west England, for which previous studies had given conflicting estimates of the probability of flooding. The novel diagnostics suggest that this discrepancy may be due to the weak dependence at extreme levels between wave periods and both wave heights and still water levels. The multivariate Gaussian tail model is shown to resolve the conflict and to offer a convincing description of the extremal sea-state process at Newlyn.

Sequential predictions of menstrual cycle lengths

Forecasting the length of the menstrual cycle and of its phases is an important problem in infertility management and natural family planning. Using repeated measurements of the length of the entire cycle and of the preovular phase provided by a large English database, we describe a Bayesian hierarchical dynamic approach to the problem. A state-space process is used to model the temporal behavior of the series of lengths for each woman. The individual processes are then embedded into a multivariate system through a Bayesian hierarchy in which model parameters are allowed to vary across subjects according to a specified probability distribution. The most interesting features of the suggested method are (a) it takes into account explicitly the temporal nature of the available data and (b) if combined with a fecundability model, it can be used to forecast the probability of conception in future cycles as a function of any intercourse behavior.

Use of a toxicokinetic model in the analysis of cancer mortality in relation to the estimated absorbed dose of dioxin (2,3,7,8-tetrachlorodibenzo-p-dioxin, TCDD)

We performed an analysis of All cancer and Lung cancer mortality in relation to estimated absorbed dose of dioxin (2,3,7,8-tetrachlorodibenzo-p-dioxin, TCDD) in the cohort of chemical workers at 12 US plants assembled by the US National Institute for Occupational Safety and Health (NIOSH) (n = 5172). Estimates of cumulative exposure to TCDD were based on a minimal physiologic toxicokinetic model (MPTK) that accounts for inter- and intra-individual variations in body mass index (BMI) over time. Population-level parameters related to liver elimination and background (input or concentration) of TCDD were estimated from separate data with repeated measures of serum TCDD (US Air Force Health Study). An occupational TCDD input parameter was estimated based on one-point-in-time TCDD data available for a subset (n = 253) of the NIOSH cohort. Model-based time-dependent cumulative dose estimates (area under the curve (AUC) of the lipid-adjusted serum TCDD concentration over time) were obtained for members of the full cohort with recorded body height and weight (n = 4049), as this information is required by the MPTK model to compute dose. Missing-value problems arose in the estimation of the occupational input parameter (n = 42) and in TCDD-dose calculation in the full cohort (n = 886) and they were handled with multiple imputation methods. Risk-regression analyses were based on Cox log-linear models including age at entry, year of entry and duration of employment as categorical covariates in addition to the logarithm of cumulative TCDD dose in ppt-years. Risk sets were stratified on birth cohort. Estimates of the unlagged exposure coefficient in these models were 0.1249 [95% confidence interval (CI) 0.0144, 0.2354] for All cancer and 0.2158 (95% CI 0.02376, 0.4078) for lung cancer. A 10-year lag produced an increase in the estimate for all cancer (0.1539, 95% CI 0.0387, 0.2691), whereas, the estimate for lung cancer was not affected much (0.2125, 95% CI 0.0138, 0.4112). At a dose level of 100 times the background the estimates obtained with a 10-year lag translate into a relative risk of 2.03 (95% CI 1.19-3.45) for all cancer and of 2.66 (95% CI 1.07-6.64) for lung cancer. Higher estimates of the exposure coefficients were obtained after imputation of missing values. This increase in risk seemed due to the inclusion of short-term workers, who may exhibit a higher mortality for reasons other than dioxin exposure.

A sufficiency property arising from the characterization of extremes of Markov chains

At extreme levels, it is known that for a particular choice of marginal distribution, transitions of a Markov chain behave like a random walk. For a broad class of Markov chains, we give a characterization for the step length density of the limiting random walk, which leads to an interesting sufficiency property. This representation also leads us to propose a new technique for kernel density estimation for this class of models.

Uncertainty in estimating exposure using a toxicokinetic model. The example of 2,3,7,8-tetrachlorodibenzo-p-dioxin.

Abstract: This paper deals with sources of uncertainty in the use of a minimal physiological toxicokinetic model to obtain dose estimates for a dose‐response analysis of cancer in an occupational cohort. Toxicokinetic models make it possible to construct exposure parameters that are more closely related to the individual dose than traditional measures of exposures to toxic agents. However, the process introduces a wide array of sources of uncertainty. Selecting a model structure to describe the kinetics of a toxic agent implies necessarily making simplifications and assumptions that influence the range of applicability of the model. Once a model has been selected, the value of certain model parameters (constants) must be assigned, for example, from anthropometric data. The question then arises of how sensitive the model predictions are to variations in the values of these constants. Other model parameters, typically those describing the kinetics of the agent, are next estimated from actual data. There may be limitations in the data concerning, for example, sparseness (too few observations per subject) or missing values. The methods used for parameter estimation carry their own set of assumptions that need to be appropriate to the situation at hand. In summary, the dioxin example is used to characterize the sources of uncertainty at different levels, such as model structure, methods and data used for parameter estimation, estimation of occupational exposure, and imputation of missing values in exposure indices derived from the kinetic model.

The status and the actions to implement the Kyoto Protocol in Italy: problems and perspectives

In this article, the authors point out main problems and perspectives related to the implementation of the Kyoto Protocol in Italy. After a brief and general introduction for explaining main concerns and institutional frameworks involved in the development of international procedures suitable to deal with the problem at a global level, the status of Italian actions in the different economic activities is presented. Finally, without any pretension to be exhaustive and with the appropriate cautions related to the number of available figures, statistical predictions are presented for Italy that highlight the main trends in the emissions of the effluents considered in the Kyoto Protocol.